Imatinib Mesylate or Dasatinib in Treating Patients With Previously Untreated Chronic Phase Chronic Myelogenous Leukemia

Inclusion Criteria: * Patients must have CML in chronic phase based on bone marrow aspiration and biopsy and peripheral blood counts obtained within 28 days before registration * Patients must be registered on this study within 180 days after the date of first being diagnosed with CML, based on a cytogenetic or molecular analysis of peripheral blood or bone marrow, demonstrating the presence of the Philadelphia chromosome or variants of the (9;22) translocation or testing positive for Bcr-Abl by real time-polymerase chain reaction (RT-PCR); patients may have secondary chromosomal abnormalities in addition to the Philadelphia chromosome and remain eligible * Patients must have Zubrod performance status 0-2 * Patients must not have received prior treatment for CML with the exception of hydroxyurea and/or anagrelide * Patients must not have received any prior chemotherapy regimen for peripheral blood stem cell mobilization; (prior collection of unmobilized peripheral blood stem cells is permitted) * Serum bilirubin =\< 2.0 x the institutional upper limit of the normal (IULN) * Serum glutamic oxaloacetic transaminase (SGOT)/aspartate aminotransferase (AST) or serum glutamate pyruvate transaminase (SGPT)/alanine transaminase (ALT) =\< 2.0 x the IULN * Patients (Southwestern Oncology Group \[SWOG\] institutions only) must be registered on SWOG-9007, "Cytogenetic Studies in Leukemia Patients;" collection of the pretreatment bone marrow specimen must be completed within 28 days prior to registration; the pretreatment bone marrow specimen must be submitted to an approved Southwest Oncology Group Cytogenetics Laboratory for cytogenetic analysis as described, and an aliquot of the bone marrow (or peripheral blood if the marrow aspiration is a dry tap) must be submitted for fluorescent in situ hybridization (FISH) analysis; note that protocol SWOG-9007 also requires submission of bone marrow specimens at the time of progression to the accelerated or blastic phase of CML or the loss of complete hematologic response and every six months while the patient is on this study * Patients (SWOG institutions only) must be registered on S9910, "Leukemia Centralized Reference Laboratories and Tissue Repositories, Ancillary;" specimens of peripheral blood and bone marrow must be submitted to the Southwest Oncology Group Lymphoid Leukemia and CML Centralized Laboratory in Seattle, Washington; collection of pretreatment blood and marrow specimens must be completed within 28 days prior to registration; note that this study also requires submission of peripheral blood and bone marrow specimens at various times after entry into the study * Patients must not have undergone major surgery within 28 days before registration, and must have fully recovered from any other prior major surgery * Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method * No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, cervical carcinoma in situ, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease-free for five years * Patients must have an electrocardiogram (ECG) within 42 days prior to registration, and must not have any of the following cardiac symptoms prior to entry on study: * Uncontrolled angina, congestive heart failure or myocardial infarction (MI) within 6 months of study entry * Diagnosed or suspected congenital long QT syndrome * Any history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or Torsades de pointes) * Prolonged corrected QT (QTc) interval on pre-entry electrocardiogram (\> 450 msec) * Uncontrolled hypertension * Patients must not have a history of significant bleeding disorder unrelated to cancer, including: * Congenital bleeding disorders (e.g., von Willebrand's disease) * Acquired bleeding disorder within one year (e.g., acquired anti-factor VIII antibodies) * All patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines * At the time of patient registration, the treating institution's name and identification (ID) number must be provided to the Data Operations Center in Seattle in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered into the data base